Exponential mixing of all orders on Kähler manifolds : (quasi-)plurisubharmonic observables

Let f be a holomorphic automorphism of a compact Kähler manifold with simple action on cohomology and µ its unique measure of maximal entropy. We prove that µ is exponentially mixing of all orders for all d.s.h. observables, i.e., functions that are locally differences of plurisubharmonic functions. As a consequence, every d.s.h. observable satisfies the central limit theorem with respect to µ.